Snowmobiles and jet-driven watercraft usually have a seat that is straddled by the driver. The seat on such vehicles is usually secured directly to the chassis of the vehicle. On a snowmobile, there may be a suspension arrangement that connects to the drive track and the skis on the front of the snowmobile to absorb some of the shocks and bumps experienced by the snowmobile in use. However, the seat of the snowmobile is usually secured directly to the chassis or frame of the snowmobile and it is only the padding of the seat that prevents the effects of bumps and other shocks from being transferred from the snowmobile frame to the rider position on the seat. There have been some attempts to provide suspension devices between the frame of the snowmobile and the seat on the snowmobile as shown in U.S. Pat. No. 5,613,570. These prior art suspension systems for a snowmobile seat have proved less than satisfactory. A vehicle, such as a snowmobile, can encounter extremely irregular terrain and frequently the changes in the terrain are not clearly visible through the snow. This can result in a wide range of bumps and shocks that must be accommodated by the suspension system for the seat. The bumps can range from relatively minor to very severe such as when the snowmobile lands after being airborne. The prior art suspension systems for seats of the snowmobile have not adequately dealt with the wide range of conditions that are experienced by the snowmobile during use. If the spring means of the seat suspension are designed to accommodate the large bumps and shocks that are experienced by the snowmobile, the springs are so stiff that they do not provide any meaningful suspension when the snowmobile encounters smaller bumps or irregularities. Accordingly, the seat suspension is not functioning to provide any meaningful suspension activity over these smaller bumps or shocks. If the springs are sized to accommodate the smaller bumps or irregularities that the snowmobile encounters, the springs are not adequate to handle the large bumps or irregularities encountered during the operation of the snowmobile. With springs that are not adequate for the larger bumps, the seat will bottom out or strike the chassis of the snowmobile when the large bumps are encountered and this significantly negates any advantage provided by the seat suspension. In fact, if the seat bottoms out against the frame or chassis of the snowmobile, this may create a very unsettling condition for the operator of the snowmobile and makes the snowmobile more difficult to operate for the driver.